Term Rewriting System R:
[x, y, z]
f(f(f(a, x), y), z) -> f(f(x, z), f(y, z))
f(f(b, x), y) -> x
f(c, y) -> y

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

F(f(f(a, x), y), z) -> F(f(x, z), f(y, z))
F(f(f(a, x), y), z) -> F(x, z)
F(f(f(a, x), y), z) -> F(y, z)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Non Termination

Dependency Pairs:

F(f(f(a, x), y), z) -> F(y, z)
F(f(f(a, x), y), z) -> F(x, z)
F(f(f(a, x), y), z) -> F(f(x, z), f(y, z))

Rules:

f(f(f(a, x), y), z) -> f(f(x, z), f(y, z))
f(f(b, x), y) -> x
f(c, y) -> y

Strategy:

innermost

Found an infinite P-chain over R:
P =

F(f(f(a, x), y), z) -> F(y, z)
F(f(f(a, x), y), z) -> F(x, z)
F(f(f(a, x), y), z) -> F(f(x, z), f(y, z))

R =

f(f(f(a, x), y), z) -> f(f(x, z), f(y, z))
f(f(b, x), y) -> x
f(c, y) -> y

s = F(f(f(a, f(a, x'''')), c), f(c, f(f(a, x''''), f(f(a, f(a, x'''')), c))))
evaluates to t =F(f(f(a, f(a, x'''')), c), f(c, f(f(a, x''''), f(f(a, f(a, x'''')), c))))

Thus, s starts an infinite chain.

Innermost Non-Termination of R could be shown.
Duration:
0:00 minutes